This means that it isn't possible to capture both the zenith and nadir
with a single row of shots - This would require both a lens with an angle
of view greater than 180 degrees and careful levelling of the camera.

However this isn't a problem if you manually tilt the camera up and down while capturing
the sequence, something that is hard with a tripod, but easy with a handheld
philopod. So the first and third shots are tilted down by 10 degrees or
so, capturing the nadir, and the second and fourth shots are tilted up 10
degrees to capture the zenith:

So when stitched together, there is a good amount of coverage of the entire scene:

One problem with this system is that the stitching seams run directly through
both the zenith and nadir. This causes problems for blending tools such as
enblend, since they have no idea that this is an
equirectangular image and get the seams wrong as a
result. It is also difficult to hand retouch the poles for the same reasons.

The solution is to stitch the panorama in different orientation with all the
seams in the middle of the output where the equirectangular distortion is
minimal:

This is very quick and easy to do, just adjust the anchor image so the weight at the end
of the philopod is in the centre of the output. This has another useful advantage -
The panorama is now completely self-levelling since the philopod forms a plum-line, so
there is no need to set vertical control points to level the scene.

The final step is to flip the panorama back to a level orientation, just load
it into one of the GUI front-ends as an equirectangular input image with a pitch of
-90 degrees and stitch. Here is the result as a QTVR:

Notes

This technique isn't limited to four shots with an up, down, up, down sequence. Lenses
with a narrower angle of view or difficult scenes might require six shots at 60 degree
rotation.

In this six shot case, the obvious sequence of up, down, up, down, up, down generates
complex seams that are difficult to disentangle, I get better results with an
up, down, down, up, down, down sequence instead.